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Measurement error in FRC/NCREIF returns on real estate.

I. Introduction

The problem of measurement error in the Frank Russell Company/National Council of Real Estate Investment Fiduciaries (FRC/NCREIF) real estate returns series is quite clearly an important one. This returns series purports to track the performance of real estate investments made by large institutional investors. But, in practice, measurement errors are likely to occur since the returns are based on appraised values, not on market prices. One recent contribution, that of Geltner [3] and Giliberto [4], suggests that the problem of measurement error in the FRC/NCREIF returns series is apt to make it appear as if real estate offers higher returns and lower portfolio risk than stocks or bonds. Geltner [3] and Giliberto [4] propose testing for the existence of measurement errors in the FRC/NCREIF returns series by measuring the extent to which appraised values depart significantly from market prices.

A more practical test procedure is to treat the problem as an errors in variables problem and specify an inventory demand model for commercial real estate put in place in which the FRC/NCREIF returns series, as a proxy for the "true" return on real estate, shows up as an explanatory variable. The appeal of this estimation procedure lies in the fact that, although it is in the spirit of Geltner [3] and Giliberto [4], it does not require the analyst to postulate how changes in appraised values are related to changes in market values; rather, the idea is to evaluate the measurement errors in the FRC/NCREIF returns series by assuming that the relationship between the inventory demand for commercial real estate put in place and the return on real estate is a deterministic one, and that the only reason why we observe some unaccounted for variation in the dependent variable is that we do not have exact measurements of the inventory demand for commercial real estate put in place or the return on real estate. For suitably defined parameter estimates, the ratio of the asymptotic error variance in the FRC/NCREIF returns series to the true return variance on real estate can then be obtained by separating out the effects of errors associated with the regression model and measurement errors. It then goes without saying that the larger the variance of the measurement error in the FRC/NCREIF returns series relative to the variance of the true returns on real estate, the less reliable the FRC/NCREIF returns series is in assessing the performance of real estate investments relative to stocks and bonds. Conversely, when the proportion of the variance in the FRC/NCREIF returns series to the variance in the true returns on real estate is small, the theoretical returns on real estate and the FRC/NCREIF returns series may be close enough that the difference is of no real consequence in assessing the diversification advantage of real estate. The problem becomes crucial only when the variance of the measurement error in the FRC/NCREIF returns series is of such size that real estate is seen as offering an attractive diversification opportunity for those invested in stocks and bonds when stocks and bonds are, in fact, clearly superior.

II. An Errors in Variables Approach to Estimate the Measurement Error in the FRC/NCREIF Returns Series

To begin with, consider a standard flexible accelerator-inventory model:
[I.sub.t] - [I.sub.t-1] = [theta]([I.sub.t] - [I.sub.t-1]), (1)
[Mathematical Expression Omitted] (2)


where

I = the value of commercial real estate put in place at the end of period t, GNP = aggregate gross national product, UNEMP = unemployment rate, r = the actual (before-tax) return on real estate, [delta] = the risk-free (before-tax) rate of return,

and where O < [theta] < 1. The tilde over [I.sub.t] denotes expected quantities. The model contained in (1) and (2) assumes that investors possess a target value of commercial real estate put in place and that construction put in place only gradually adjusts toward the target level. The model also assumes that an increase in the rate of return on commercial real estate will tempt investors to put more of their assets into commercial real estate and less into other investments. The variable [delta].sub.t] is entered separately in order to capture the cost of holding inventories. Models of this nature have been estimated for manufacturing inventories and consumer durables by Blinder [1], and Maccini and Rossana [8], among others.

Estimation of (1) and (2) tends to be problematic since, instead of [I.sub.t], and [r.sub.t], we normally observe [I.sub.t.sup.*] and [r.sub.t.sup.*]. Thus, a typical reduced-form regression equation - ignoring, for the moment, the variables [GNP.sub.t], [UNEMP.sub.t], [delta].sub.t], and [I.sub.t-i] - might then be

[I.sub.t.sup.*] = [alpha] + [[beta][r.sub.t.sup.*] + [epsilon].sub.t.sup.*]

(3) where

[I.sub.t.sup.*] = [I.sub.t] + [v.sub.t] (4)

[r.sub.t.sup.*] = [r.sub.t] + [xi].sub.t] (5)

[epsilon].sub.t.sup.*] = [epsilon].sub.t] + [v.sub.t] - [beta] [xi].sub.t]

(6) and where [epsilon].sub.t] is a disturbance term with mean zero and [sigma].sub.[epsilon].sup.2] in the "true" regression model [I.sub.t] = [alpha] + [beta][r.sub.t] + [epsilon].sub.t], and [v.sub.t] and [xi].sub.t] represent the errors in measuring the t th value of I and of r.(1) Asterisks in this case denote the observable values of the variables.

The error terms [v.sub.t] and [xi].sub.t] are distributed
 [v.sub.t] N(O, [sigma].sub.v.sup.2], (7)
 [xi].sub.t] N(O, [sigma].sub.[sigma].sup.2]; (8)


where E([.sub.i][v.sub.j]) = O, E([xi].sub.i][xi].sub.j] = O for i # j, and E([v.sub.i][xi].sub.j]) = O, E([v..sub.i][epsilon].sub.j]) = O, and E([xi].sub.i][epsilon].sub.j]) = O. Assumptions (7) and (8) state that each error is a random variable with zero mean and constant variance. The assumptions E([v.sub.i][v.sub.j]) = O, E([xi].sub.i][xi].sub.j]) = O for i # j rule out situations in which the errors are autoregressive and E([v.sub.i][xi].sub.j]) = O, E([v.sub.i][epsilon].sub.j]) = O, and E([xi].sub.i][epsilon].sub.j]) = O state that the errors are unrelated to each other. Of course, the real problem is not that [I.sub.t] is measured with error, but rather it is because [r.sub.t] is measured with error.

When ordinary least squares (OLS) is applied to equation (3), the OLS estimators of [alpha] and [beta] are

plim [alpha] = [I.sup.-*] - [beta]([sigma].sub.r.sup.2]/([sigma].sub.r.sup.2] + [sigma].sub.[xi].sup.2]))[r.sup.-*] (9)

plim [beta] = [beta]([sigma].sub.r.sup.2]/([sigma].sub.r.sup.2] + [sigma].sub.[xi].sup.2])) (10)

where the expression plim refers to the probability limit as n - [infinity] From (10), it is easily seen that the direct OLS estimate of [beta] is biased. Moreover, the bias does not decrease with the sample size. The magnitude of the bias in [beta] is determined by the ratio of the error variance [sigma].sub.[xi].sup.2] to [sigma].sub.r.sup.2].(2)

An alternative method of estimating equation (3) which will yield unbiased estimates and help us determine the size of the measurement error of [r.sub.t.sup.*] is to apply Wald's method of group averages. Wald's method of group averages requires ordering the observed pairs (r.sub.t.sup.*], [I.sub.t.sup.*]) by the magnitude of the [r.sub.t.sup.*] so that

[r.sub.1.sup.*] [less than or equal to] [r.sub.2.sup.*] ... [less than or equal] [r.sub.n.sup.*]. (11)

The pairs are then divided into three groups of approximately equal size, and the instrumental variable [Z.sub.i] is defined such that

[Z.sub.i] = -1 if i belongs to the 1st group,
 0 if i belongs to the 2nd group, (12)
 1 if i belongs to the 3rd group.


Next, an instrumental-variables equation for [r.sub.t.sup.*] is estimated by regressing [r.sub.t.sup.*] on [Z.sub.i], that is,

[r.sub.t.sup.*] = [gamma].sub.o] + [gamma].sub.1][Z.sub.i] + [mu].sub.t]

(13) where [mu] is an error term. From (13), the fitted values of the dependent variable [r.sub.t.sup.*] are determined. The fitted values [r.sub.sup.*] will by construction be independent of the errors terms in equation (3), as well as the errors of measurement of both variables. Thus, using [r.sub.t.sup.*] allows one to construct a variable which is linearly related to [r.sub.t.sup.*] and which is purged of any correlation with the error term in the reduced-form inventory equation. The fitted variable [r.sub.t.sup.*] can then be used to replace [r.sub.t.sup.*] in equation (3) and ordinary least squares utilized to estimate [alpha] and [beta]. The resulting Wald estimators of a and [beta] are

[Mathematical Expression Omitted]

From the condition that

[Mathematical Expression Omitted] (16) it follows that [alpha][psi] and [beta][psi] are consistent estimators of [alpha] and [beta].

From (10) and (15), a reasonable estimate of the ratio of the asymptotic error variance [sigma].sub.[xi].sup.2] to [sigma].sub.r.sup.2] is

[Mathematical Expression Omitted] (17)

Note that when [beta] is a good approximation to [beta][psi], then [beta][psi]/[beta] and [sigma].sub.r.sup.2] are approximately unity and zero, respectively. That is, only when [beta][psi] [much greater than] [beta] will [sigma].sub.[xi].sup.2]/[sigma].sub.r.sup.2] [much greater than] 0. Next, using equation (5), one obtains

[Mathematical Expression Omitted] (18) where [sigma].sub.r*.sup.2] is the asymptotic variance of the observed return on real estate. Equation (18) therefore gives a straightforward method of estimating the error variance in the FRC/NCREIF returns series. Instead of postulating how changes in appraised values are related to changes in market values, the idea is to evaluate the measurement error in the FRC/NCREIF returns series by estimating an equation similar to (3) using Wald's method of group averages. Then, given an estimate of [sigma].sub.r*.sub.2] and the results from equation (17), it is quite easy to determine the true variance of the FRC/NCREIF returns series: it is simply [sigma].sub.r*.sup.2] from equation (18).

III. Some Empirical Results

In this section, the results of estimating the flexible accelerator-inventory model given in (1) and (2) using Wald's method of estimation are reported. Substituting (2) into (1) yields the reduced-form equation

[Mathematical Expression Omitted] (19) where [b'.sub.i] = [theta][b'.sub.o], and [I.sub.t.sup.*] = [I.sub.t] + [v.sub.t] and [r.sub.t.sup.*] = [r.sub.t] + [xi].sub.t].

Equation (19) is estimated separately for office and industrial buildings. The demand for office buildings includes the demand for single-tenant buildings, smaller multi-tenant mid-rise structures, and high-rise central business district buildings. The demand for industrial buildings includes warehouses, service center buildings, high tech buildings, office-showroom facilities, and manufacturing facilities. Apart from the obvious benefits of examining data for more than one type of real estate, office and industrial buildings were chosen since FRC/NCREIF returns data on office and industrial real estate investments were readily accessible.(3)

Figure 1 illustrates the marked difference in the FRC/NCREIF returns on office buildings during the 1980:I-1981:IV and 1986:I-1990:II periods. In the former period, the FRC/NCREIF returns on office buildings averaged 5.43 percent per quarter; in the latter period the rate of return averaged 0.64 percent per quarter. The downward drift in the FRC/NCREIF returns series begins around 1985:I-1986:IV. Figure I also shows the total U.S. office construction put in place during the 1980:I-1990:II period. The series was taken from Citibase and represents the annual rate of office construction put in place expressed in billions of 1982 dollars. Other data needed to analyze commercial real estate markets appear to be missing or available only at infrequent intervals.(4)

As can be seen in Figure 1, the value of office construction put in place was much greater in 1985:I-1986:IV and 1988:I-1989:IV than in the early 1980s. It would appear from these data that there is no simple relationship between office construction put in place and the FRC/NCREIF return series. Rosen [9], and Wheaton [12] have noted that modeling the office construction cycle by traditional equilibrium models is difficult because office construction is so variable.(5)

The time series for quarterly FRC/NCREIF returns on industrial properties and the annual rate of industrial construction put in place are shown in Figure 2. The data on the annual rate of industrial real estate construction put in place were derived from Citibase. The FRC/NCREIF returns series fluctuates considerably around 2.75 percent per quarter during the 1980:I-1990:II period. Also note that there is an enormous difference in total industrial construction put in place in the interim period between 1981:I-1982:IV and 1989:I-1990:II. In the earlier period, the annual rate of industrial construction put in place averaged $52.7 billion (constant 1982 dollars); in the latter period the annual rate of industrial construction put in place averaged $52.5 billion. During the 1983:I-1988:IV interim period, the annual rate of industrial construction put in place averaged $37.9 billion. Wheaton and Torto [13] argue that the industrial real estate market in the United States is substantially different from that for office buildings since most industrial properties are owner-occupied and a majority of the rented structures have a single, long-tertn tenant.

The Results for Office Buildings

The results of estimating equation (19) for office buildings are reported in columns (1) and (2) of Table I. Columns (1) and (2) are for the 1980:II-1990:II period and differ only in that the parameters in column (1) were estimated using OLS while the parameters in column (2) were estimated using Wald's method of group averages. The reason for lagging [r.sub.t.sup.*] one period is based on the overall statistical fit of the equation and its inability to pick up turns in the value of office construction put in place. Lagging [r.sub.t.sup.*] one period also accounts for the fact that the FRC/NCREIF returns series is very slow to adjust during periods of declining values [6].

The OLS estimates verify the expected results that the least squares estimator of [b'.sub.3] is closer to zero than the Wald estimator of [b'.sub.3]. The OLS estimator of [b'.sub.3] for office buildings is .856 with a t-statistic of t = 1.50. By contrast, the Wald estimator of [b'.sub.3] is 2.697 and, with a t-statistic of t = 2.43, it is significantly different from zero at the 0.05 level. The speed of adjustment coefficient [theta] suggests that the influence of a change in [Mathematical Expression Omitted] is spread over several periods of time. The adjustment coefficient is between .494 and .583.

The [R.sup.2s] for both equations are virtually identical. The reported F-values are 26.42 and 29.80. Other things being equal, one would expect a strong statistical relationship between office construction put in place and the FRC/NCREIF returns series to result in a large F-value. Both F-tests are able to reject the null hypothesis that there is no relationship between [I.subt.sup.*] and [r.sub.t.sup.*] at the 0.05 level.

[TABULAR DATA OMITTED]

An estimate of the ratio of the asymptotic error variance [[sigma].sub.[xi].sup.2] to [[sigma].sub.r.sup.2] can be determined from the empirical results in Table I. If one assumes the behavioral characteristics given in equations (7) and (8), then [Mathematical Expression Omitted] where [b.sub.3.sup.[psi]] is the Wald estimator of [b'.sub.3]. The last row in Table I sets out the estimated ratio of the asymptotic error variance [[sigma].sub.[xi].sup.2] to [[sigma].sub.r.sup.2]. With reasonable values of [b.sub.3.sup.[psi]] = 2.697 and [Mathematical Expression Omitted] for office buildings.

The Results for Industrial Buildings

For industrial buildings, equation (19) was estimated over the same sample period as for the office market. The results are reported in columns (3) and (4) of Table I. The lag structure for [r.sub.t.sup.*] was found empirically.

The coefficients on [r.sub.t-1.sup.*] are consistent with theory. As was the case for the office market, the OLS estimate of [b'.sub.3] is closer to zero than Wald's estimator of [b'.sub.3]. The OLS estimator of [b'.sub.3] for industrial buildings is .495 with a t-statistic of t = 1.00, while the Wald estimator of [b'.sub.3] is 1.586 and, with a t-statistic of t = 2.15, it is significantly different from zero at the 0.05 level. Note that the speed of adjustment coefficient [theta] is much lower for industrial buildings than office buildings. For industrial buildings, the value of [theta] is between .109 and .311.

The respective values for the [R.sup.2]s are .93 for the OLS equation and 0.87 for the Wald model. A F-test at the 5 percent level of significance rejects the null hypothesis that there is no relationship between industrial construction put in place and return on industrial real estate for both models.

The message in Table I is that the ratio of the asymptotic error variance [[sigma].sub.[xi].sup.2] to [[sigma].sub.r.sup.2] for industrial real estate is similar to the ratio of the asymptotic error variance for office buildings. With reasonable values of [b.sub.3.sup.[psi]] = 1.586 and [b'.sub.3] = .495, the value of [Mathematical Expression Omitted] for industrial buildings is 2.20. It would therefore seem that, as far as measurement errors in the FRC/NCREIF returns series are concerned, there is identical appraisal bias in both the office and industrial returns series.

Table II. Annualized Returns on Selected Asset Classes with and without Corrections for Measurement Errors in the FRC/NCREIF Returns Series, 1981:IV-1986:I
Asset Class Mean Return, % Standard Deviation Reward-to-volatilit
y
Stocks(a) 20.46 14.64 1.40
Bonds(b) 19.06 7.94 2.40


Real Estate:(c)
All Properties 10.84 1.76 6.16
Office 11.14 5.36 2.07
Industrial 10.92 2.96 3.69


Notes:

a. Total returns earned by the S&P 500. b. Returns calculated from the Salomon Brothers Broad Investment-grade Bond Index. c. Returns calculated from the FRC/NCREIF retums series.

Implications for Real Estate Investors

In several recent studies by Firstenberg, Ross, and Zisler [2], Hartzell, Hekman, and Miles [5], and others, it is concluded that commercial real estate tends to outperform investments in stocks and bonds, with higher returns and greater diversification benefits for given risk levels. These findings, which are based on mixed-asset portfolio simulations using the FRC/NCREIF returns series, are admittedly biased since the potential misestimation of the appreciation component of the FRC/NCREIF returns series may lead to a substantial underrepresentation of volatility for real estate returns. It is therefore useful to ask the question, How would these findings differ if a correction were made to the FRC/NCREIF returns series for measurement errors?

To answer this question, let us consider one of the holding periods most often used as justification for abnormal performance of real estate, 1981:IV-1986:I. During this period, annualized FRC/NCREIF returns on office and industrial buildings averaged 11.14 and 10.92 percent, respectively, and the corresponding standard deviations were 5.36 and 2.96 (see Table II). This yields a Sharpe's reward-to-volatility measure, which is often used to evaluate the financial performance of an asset, of 2.08 for office buildings and 3.69 for industrial buildings.(6) By comparison, the Sharpe's reward-to-volatility measure for stocks during this period was 1.40.(7) Based on these approximate but rudimentary comparisons, we are therefore led to believe that real estate offers investors higher returns for a given level of risk. But, if we truly are to ascertain whether real estate offers higher returns per unit of risk than stocks, we must adjust the FRC/NCREIF returns series for potential measurement errors.

With values of [[sigma].sub.r] in the range between 6 and 8 percentage points, the best estimate of [[sigma].sub.r.sup.*] for office buildings is, using equation (18) and the results given in Table I, between 12.9 and 17.2 percentage points.(8) Likewise, with values of [[sigma].sub.r] in the range between 4 and 5 percentage points for industrial buildings, the best estimate of [[sigma].sub.r.sup.*] for industrial properties is between 8.8 and 11.0 percentage points. By comparison, standard deviations of stocks and bond returns are usually around 8 to 15 percentage points. Values of [[sigma.sub.r] are from Webb, Miles, and Guilkey [11].

The higher [[sigma.sub.r.sup.*]'s imply lower Sharpe's reward-to-volatility measures for both office and industrial buildings. The new Sharpe ratios for office and industrial buildings are between .65 and 1.24. Thus it would appear that the simple reward-to-volatility measures for real estate are not nearly as attractive as they would seem to be at first blush. This might explain why institutional involvement in real estate has traditionally been so minuscule.

IV. Summary

The analysis of the inventory demand for commercial real estate put in place suggests that significant measurement errors are likely to be present in die FRC/NCREIF returns series. This is an important observation, because, as pointed out at the outset, measurement errors in the FRC/NCREIF returns series may lead to erroneous conclusions regarding whether real estate offers higher returns and lower portfolio risk than stocks or bonds, or whether real estate offers an attractive diversification opportunity for those invested in stocks and bonds. The analysis also suggests that the proportion of the variance of the measurement error to the variance of the "true" return on real estate is essentially identical in both the FRC/NCREIF office and industrial returns series.

References

[1.] Blinder, Alan S., "Retail Inventory Behavior and Business Fluctuations." Brookings Papers on Economic Activity, 2,1981, 443-505. [2.] Firstenberg, Paul M., Stephen A. Ross, and Randall C. Zisler, "Real Estate: The Whole Story." Journal of Portfolio Management, Spring 1988, 22-34. [3.] Geltner, David, "Bias in Appraisal-based Returns." Journal of the American Real Estate and Urban Economics Association, Fall 1989, 338-52. [4.] Giliberto, Michael, "A Note on the Use of Appraisal Data in Indexes of Performance Measurement." Journal of the American Real Estate and Urban Economics Association, Spring 1988, 77-83. [5.] Hartzell, David J., John Hekman, and Mike Miles, "Diversification Categories in Investment Real Estate." Journal of the American Real Estate and Urban Economics Association, Spring 1986,230-54. [6.] Hendershott, Patric H. and Edward J. Kane. "Office Market Values During the Past Decade: How Distorted Have Appraisals Been?" Working paper, Ohio State University, 1992. [7.] Kling, John L. and Thomas E. McCue, "Office Building Investment and the Macroeconomy: Empirical Evidence, 1973-1985." Journal of the American Real Estate and Urban Economics Association, Fall 1987, 234-55. [8.] Maccini, Louis J. and Robert J. Rossana, "Joint Production, Quasi-fixed Factors of Production, and Investment in Finished Goods Inventories." Journal of Money, Credit, and Banking, May 1984, 218-36. [9.] Rosen, Kenneth T., "Towards a Model of the Office Building Sector." Journal of the American Real Estate and Urban Economics Association, Fall 1984, 261-69. [10.] Theil, Henri. Principles of Econometries. New York: John Wiley & Sons, Inc., 1971. [11.] Webb, R. Brian, Mike Miles and David Guilkey, "Transactions-driven Commercial Real Estate Returns: The Panacea to Asset Allocation Models?" Journal of the American Real Estate and Urban Economics Association, Summer 1992, 325-40. [12.] Wheaton, William C., "The Cyclic Behavior of the National Office Market." Journal of the American Real Estate and Urban Economics Association, Winter 1987, 281-99. [13.] ----- and Raymond G. Torto, "An Investment Model of the Demand and Supply for Industrial Real Estate." Journal of the American Real Estate and Urban Economics Association, Winter 1990, 530-47.

(1.) The expression for [epsilon].sub.t.sup.*] can be derived by substituting [I.sub.t] = [I.sub.t.sup.*] - [v.sub.t] and [r.sub.t] = [r.sub.t.sup.*] - [xi].sub.t] into the regression equation [I.sub.t] = [alpha] + [beta][r.sub.t] + [epsilon].sub.t] and rewriting the model as ([I.sub.t.sup.*] - [v.sub.t] = [alpha] + [beta]([r.sub.t.sup.*] - [xi].sub.t]) + [epsilon].sub.t]. This is equivalent to equation (3) in the text where [epsilon].sub.t.sup.*] = [epsilon].sub.t] + [v.sub.t] - [beta][xi].sub.t]. (2.) Specific details about the measurement bias in [beta] are given in Theil [10]. (3.) Office buildings are often classed by construction type, from class A through D, with the premium buildings being classified as class A. Likewise, industrial buildings are often classified by construction type and as being with or without dock-height loading and as being rail or non-rail served. Returns data, however, are not available at this level of disaggregation. (4.) The construction put in place series is derived directly from the Bureau of the Census, Construction Progress Reporting Survey and represents the best available measure of changes in inventories under construction for commercial real estate. Estimates of value put in place are based on an independent systematic sample of roughly 4,900 projects in the U.S. at any one time. Once a project is selected, monthly construction progress reports are requested from the owner until the project is complete. (5.) Little is known about the office construction cycle. Kling and McCue [7] find that office building investment is responsive to a decline in nominal interest rates, and that shocks in output and aggregate prices affect office construction only with a lag. Their model ignores the fundamental relationship given in (1) and (2). (6.) Sharpe's reward-to-volatility measure is total return divided by standard deviation of return. (7.) Total returns earned by the S&P 500 during the 1981:IV-1986:I averaged 20.46 percent per year with a standard deviation of 14.64. Thus, the Sharpe's reward-to-volatility measure is 20.46/14.64 = 1.40. (8.) Note that from equation (18), [Mathematical Expression Omitted] This implies that [Mathematical Expression Omitted] is estimated from Table I.
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Author:Shilling, James D.
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Date:Jul 1, 1993
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